IP AMaths Notes (Upper Sec, Year 3-4): 16) Applications of Differentiation

Study guide08 Nov 2025, 00:00 ZUpdated 30 Nov 2025

Tangents, normals, stationary points, and related rates in IP AMaths.

Download PDF Join our Telegram study group

Q: What does IP AMaths Notes (Upper Sec, Year 3-4): 16) Applications of Differentiation cover?
A: Tangents, normals, stationary points, and related rates in IP AMaths.

Apply derivatives to describe gradient, optimise functions, and connect rates.

Keep the full topic roadmap handy via our IP Maths tuition hub so you can jump into related drills, quizzes, or diagnostics as you move through these notes.

New to the Integrated Programme? Start with What is IP? | Browse all free IP notes.

Applications here match the SEAB GCE O-Level Additional Mathematics (4049) requirements: tangents/normals, stationary points with second-derivative tests or sign charts, simple optimisation, and related rates in radians.

Status: SEAB O-Level Additional Mathematics 4049 syllabus (exams from 2025) checked 2025-11-30 - scope unchanged; remains the reference for this note.

The core idea is simple: Applications of differentiation turn gradients into decisions.

Use it as a working check: Use derivatives to find tangent gradients, normal gradients, stationary points, and related rates. Always classify stationary points before naming a maximum or minimum.

Then go one layer deeper: Example: if y' = 0 at x = 3, test the second derivative or a sign chart before calling it a turning point. If y'' is positive, it is a local minimum.

Choosing the application route

Start by deciding what the derivative represents in the question. A derivative can be a gradient, a condition for a stationary point, or a link between changing quantities.

Question cue	First derivative step	What must be checked
Tangent at $x = a$

Reviewed by

Marcus Pang·Managing Director (Maths)

Sources

SEAB GCE O-Level Additional Mathematics (4049) syllabus (examinations from 2025)

Step	Tangent line	Normal line	Common trap
1. Differentiate	Find $y'$ .	Find $y'$ first, because the normal depends on the tangent gradient.	Starting with the reciprocal before finding the tangent gradient.
2. Substitute the contact value	Put $x = a$ into $y'$ to get $m_\text{tangent}$ .	Use the same contact point as the tangent.	Using a nearby point from the sketch instead of the exact point on the curve.
3. Find the point	Put $x = a$ into the original curve to get $(a, y(a))$ .	The normal also passes through $(a, y(a))$ .	Substituting $x = a$
4. Write the line	Use $y - y_1 = m(x - x_1)$ .	Use $m_\text{normal} = -1/m_\text{tangent}$

Setup question	What to write	Common trap
Which quantities are linked geometrically?	An equation such as area, volume, or Pythagoras that connects them.	Differentiating before removing unnecessary variables.
Which rate is given?	Mark it as a derivative with respect to time, such as $\dfrac{\mathrm{d}V}{\mathrm{d}t}$ .	Treating $\dfrac{\mathrm{d}V}{\mathrm{d}t}$ as if it were just $V$ .
Which rate is required?	Rearrange after differentiating to isolate the requested derivative.	Giving $\dfrac{\mathrm{d}V}{\mathrm{d}t}$ when the question asks for $\dfrac{\mathrm{d}h}{\mathrm{d}t}$
At what instant is the rate needed?	Substitute the instant-specific length, radius, height, or volume after differentiating.	Substituting changing values too early and turning variables into constants.

Evidence	Classification	What to write
$y'$ changes from positive to negative.	Local maximum.	The curve rises before the point and falls after it.
$y'$ changes from negative to positive.	Local minimum.	The curve falls before the point and rises after it.
$y'$ keeps the same sign on both sides.	Stationary point of inflexion.	The curve flattens but does not turn.
$y'' > 0$ at the stationary point.	Local minimum.	The curve is concave upward there.
$y'' < 0$ at the stationary point.	Local maximum.	The curve is concave downward there.
$y'' = 0$ at the stationary point.	Test is inconclusive.	Use a sign chart for $y'$ instead of guessing.

IP AMaths Notes (Upper Sec, Year 3-4): 16) Applications of Differentiation

Choosing the application route

Sources

1 Common tasks

Tangent and normal line checkpoint

Related-rates setup checkpoint

2 Worked example - Tangent and normal

3 Worked example - Optimisation

Stationary-point classification checkpoint

4 Worked example - Classifying stationary points

5 Practice Quiz

6 Try this

Related Posts